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"... Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse ..."

Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1

...ring of logics leads to a new logic where not only constructors are mixed, but proof methods are combined. Although the fibring techniques can be defined in the context of quantificational logic (cf. =-=[17]-=-), even if restricted to the propositional level (avoiding variables, terms, binding operators such as quantifiers, and the subtleties therein), ∗ This work started during a visit by Walter A. Carniel...

"... The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich ..."

The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOL-enrichments. Soundness is shown to be preserved by fibring without any further assumptions.

...we established the preservation of completeness when fibring propositional based logics endowed with an algebraic semantics. This transference result was later extended to first-order based logics in =-=[12]-=-, albeit at the expense of a quite complex semantics. It seems worthwhile to pursue also the study of fibring of higher-order based logics. Indeed, the applications that have been motivating our work ...

"... wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are n ..."

wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non{truth{functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3], everything can be smoothly lifted to the fully edged indexed case. In the sequel, AlgSig&apos; denotes the category of algebraic many sorted signatures with a distinguished sort &apos; (for formulae) and morphisms preserving it. Given one such signature , we denote by Alg() the category of {algebras and {algebra homomorphisms, and by cAlg() the class of all pairs hA; i with A a&lt;

...yfying a logic [11]. Other interesting applications of fibring, in a truth-functional setting, have been explored elsewhere and include, for instance, the interplay between modalities and quantifiers =-=[17]-=- and a treatment of partiality in the context of equational logic [6]. Moreover, and most importantly, we intend to study the extension to this general setting of the soundness and completeness preser...

"... In [6] it was shown that fibring could be used to combine institutions presented as c-parchments, and several completeness preservation results were established. However, their scope of applicability was limited to propositional-based logics. Herein, we extend these results to a broader class of ..."

In [6] it was shown that fibring could be used to combine institutions presented as c-parchments, and several completeness preservation results were established. However, their scope of applicability was limited to propositional-based logics. Herein, we extend these results to a broader class of logics, possibly including variables, terms and quantifiers.

...ER, namely, via the Project FibLog POCTI/MAT/372 39/2001 of CLC. free in a formula. The idea of making these side conditions explicit is not new [21], but the technique we shall use is improved along =-=[7, 22]-=-. This fine control of instantiations also has an impact on fibring, again characterizable by colimits. These aspects settled, we can study soundness and completeness transfer results in a broader con...

"... Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have ..."

Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semi-decidability is investigated.

...irst-order logic. Although preservation of soundness, completeness and interpolation has been already investigated in the context of propositional-based logics [29, 24, 4], first-order quantification =-=[23]-=-, higher-order quantification [6], non truth-functional semantics [3], sequent calculus and other deductive systems [17, 21], other forms of preservation are still to be fully understood, namely the o...

"... Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. Mostly, the research on fibring has focused on its model and proof-theoretic aspects, and on transference results for relevant metalogical properties. Conservati ..."

Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. Mostly, the research on fibring has focused on its model and proof-theoretic aspects, and on transference results for relevant metalogical properties. Conservativity, however, a property that lies at the very heart of the orig-inal definition of fibring, has not deserved similar attention. In this paper, we provide the first full characterization of the conserva-tivity of fibred logics, in the special case when the logics being combined do not share connectives. Namely, under this assumption, we provide necessary and sufficient conditions for a fibred logic to be a conservative extension of the logics being combined. Our characterization relies on a key technical result that provides a complete description of what follows from a set of non-mixed hypotheses in the fibred logic, in terms of the logics being combined. With such a powerful tool in hand, we also explore a semantic ap-plication. Namely, we use our key technical result to show that finite-valuedness is not preserved by fibring.

...of modal logics [23], which uses ideas from modal semantics that cannot be easily generalized. A particular form of failure of conservativity, known as the collapsing problem has also been studied in =-=[14, 8, 21, 11]-=-. However, if one considers the problem in full generality, it is immediate that a complete characterization of conservativity for fibred logics is far from obvious, even more so when the logics at ha...

"... Abstract. In [12, 16] we showed how to combine propositional BDI logics us-ing Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableau-based decision procedure for the combined/fibred logics. To achieve this end we first outline with an example two type ..."

Abstract. In [12, 16] we showed how to combine propositional BDI logics us-ing Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableau-based decision procedure for the combined/fibred logics. To achieve this end we first outline with an example two types of tableau systems, (graph &amp; path), and discuss why both are inadequate in the case of fib-ring. Having done that we show how to uniformly construct a tableau calculus for the combined logic using Governatori’s labelled tableau system KEM.

...gics being combined has been obtained [13,5,4,20,21]. Also, investigations related to using fibring as a combining technique in various domains has produced a wealth of results as found in works like =-=[8,18,22,19,6]-=-. The novelty of combining logics is the aim to develop general techniques that allow us to produce combinations of existing and well understood logics. Such general techniques are needed for formalis...

"... The method of fibring for combining logics as originally proposed by Gabbay [13, 14], includes some other methods as fusion [29] as a special case. Albeit fusion is the best developed mechanism, mainly in what concerns preservation of properties as ..."

The method of fibring for combining logics as originally proposed by Gabbay [13, 14], includes some other methods as fusion [29] as a special case. Albeit fusion is the best developed mechanism, mainly in what concerns preservation of properties as

...one of the main research trends in fibring. Although preservation of soundness and completeness has been investigated in the context of propositional-based logics [32, 28], first-order quantification =-=[27]-=-, higher-order features [9], non truth-functional semantics [4], sequent and other deduction calculi [16, 24], other forms of preservation are still to be investigated. We outline here some results on...

"... The modern view of logical reasoning as modeled by a consequence operator (instead of simply by a set of theorems) has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambig ..."

The modern view of logical reasoning as modeled by a consequence operator (instead of simply by a set of theorems) has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambiguous way, to describe several distinct modes of reasoning over the same logical language. A paradigmatic example of such a situation is ‘modal logic’, a designation which can encompass reasoning over Kripke frames, but also over Kripke models, and in any case either locally (at a fixed world) or globally (at all worlds). Herein, we adopt a novel abstract notion of logic presented as a lattice-structured hierarchy of consequence operators, and explore some common proof-theoretic and model-theoretic ways of presenting such hierarchies through a collection of meaningful examples. In order to illustrate the usefulness of the notion of hierarchical consequence operators we address a few questions in the theory of combined logics, where a suitable abstract presentation of the logics being combined is absolutely essential, and we show how to define and achieve a number of interesting preservation results for fibring, in the context of 2-hierarchies.

"... In [15, 19] we showed how to combine propositional multimodal logics using Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableau-based proof technique for the combined/fibred logics. To achieve this end we first make a comparison between two types of ..."

In [15, 19] we showed how to combine propositional multimodal logics using Gabbay’s fibring methodology. In this paper we extend the above mentioned works by providing a tableau-based proof technique for the combined/fibred logics. To achieve this end we first make a comparison between two types of tableau proof systems, (graph &amp; path), with the help of a scenario (The Friend’s Puzzle). Having done that we show how to uniformly construct a tableau calculus for the combined logic using Governatori’s labelled tableau system KEM. We conclude with a discussion on KEM’s features.

...he logics being combined has been obtained [16,4,22]. Also, investigations related to using fibring as a combining technique in various domains has produced a wealth of results as found in works like =-=[8,24,21,5]-=-. The novelty of combining logics is the aim to develop general techniques that allow us to produce combinations of existing and well understood logics. Such general techniques are needed for formalis...